📚 7E Properties of Functions | 7E 函数性质
In IB Mathematics, the topic "Properties of Functions" brings together domain, range, symmetry, monotonicity, inversion, composition, periodicity, boundedness, and asymptotes. Mastering these properties allows students to sketch, analyse, and transform functions confidently, and to see how different algebraic features shape the behaviour of a graph. This article walks through every key property with clear examples and paired bilingual explanations, directly mapped to the IB AA / AI syllabus section 7E.
在IB数学中，“函数性质”这一主题整合了定义域、值域、对称性、单调性、反函数、复合函数、周期性、有界性和渐近线等内容。掌握这些性质能够帮助学生自信地绘制、分析和变换函数，并理解代数特征如何决定图像的行为。本文逐项解析每一个关键性质，配以清晰示例和中英双语解释，完全对应 IB AA / AI 大纲第7E节。
1. Domain and Range | 定义域与值域
utput values (y-values) produced by the function. For a rational function like f(x) = 1/(x – domain is all real numbers except 2. The range is all real numbers except 0, because 1/(x – 2) never equals zero.
函数的定义域是所有使函数有意义的输入值（x 值）的集合，而值域是函数产生的所有输出值（y 值）的集合。对于有理函数 f(x) = 1/(x – 2)，分母不能为零，因此 x ≠ 2；定义域为除2以外的全体实数。值域为除0以外的全体实数，因为 1/(x – 2) 永远不等于0。
When a square root is present, such as g(x) = √x + 3, the quantity inside the root must be non‑negative, giving x + 3 ≥ 0, so the domain is x ≥ –3. The principal square root always outputs values ≥ 0, thus the range is y ≥ 0.
当出现平方根时，如 g(x) = √x + 3，根号内的式子必须非负，即 x + 3 ≥ 0，所以定义域为 x ≥ –3。主平方根总是输出 ≥ 0 的值，因此值域为 y ≥ 0。
In IB problems, you are often asked to state the domain and range using set notation, interval notation, or inequalities. Always check for restrictions such as denominators, even roots, and logarithms (argument must be positive).
在IB题目中，经常要求用集合记号、区间记号或不等式陈述定义域和值域。务必检查限制条件，例如分母、偶次方根、对数（真数必须为正）。
2. Even and Odd Functions | 奇函数与偶函数
Even functions satisfy the condition f(–x) = f(x) for all x in the domain, meaning their graphs are symmetric about the y‑axis. Typical examples are f(x) = x², f(x) = cos x, and f(x) = |x|. To prove a function is even, replace x with –x and simplify; the expression must match the original.
偶函数满足 f(–x) = f(x) 对其定义域内所有 x 成立，意味着图像关于 y 轴对称。典型的例子有 f(x) = x²、f(x) = cos x 和 f(x) = |x|。要证明一个函数是偶函数，将 x 换成 –x 并化简；表达式必须与原式一致。
Odd functions satisfy f(–x) = – ) = 1/x (for x ≠ 0). If an odd function is defined at 0, then f(0) must equal 0 because f(–0) = –f(0) ⇒ f(0) = –f(0) ⇒ 2f(0)=0.
奇函数满足 f(–x) = –f(x)。它们的图像关于原点呈 180° 旋转对称。例子包括 f(x) = x³、f(x) = sin x 和 f(x) = 1/x（x ≠ 0）。如果奇函数在 0 处有定义，则必有 f(0) = 0，因为 f(–0) = –f(0) ⇒ f(0) = –f(0) ⇒ 2f(0)=0。
Many functions are neither even nor odd; for instance, f(x) = x² + x shows f(–x) = x² – x, which is neither f(x) nor – ams often ask you to determine parity algebraically and explain the symmetry of the graph.
许多函数既不是奇函数也不是偶函数；例如 f(x) = x² + x 给出 f(–x) = x² – x，既不等于 f(x) 也不等于 –f(x)。IB 考试经常要求通过代数方法判断奇偶性并解释图像的对称性。
3. Monotonicity: Increasing and Decreasing | 单调性：递增与递减
itive slope m > 0 implies strictly increasing; negative m < 0 implies strictly decreasing.
如果对于区间内任意 x₁ < x₂ 都有 f(x₁) < f(x₂)，则函数在该区间上严格递增。从图像上看，从左向右移动时曲线上升。对于线性函数 f(x) = mx + c，正的斜率 m > 0 意味着严格递增；负的 m < 0 意味着严格递减。
Showing monotonicity for non‑linear functions usually involves the derivative f'(x). If f'(x) > 0 for all x in an interval, the function is strictly increasing there. If f'(x) < 0, it is strictly decreasing. A turning point occurs where f'(x) = 0, separating intervals of different monotonic behaviour.
证明非线性函数的单调性通常需要用到导数 f'(x)。如果在某个区间内 f'(x) > 0，函数在该区间严格递增。如果 f'(x) < 0，则严格递减。f'(x) = 0 处出现驻点，将不同单调性的区间分隔开。
In IB questions, you might analyse monotonicity for polynomials, exponentials, and trigonometric functions. It is also possible to discuss "non‑decreasing" or "non‑increasing" when the function stays flat on some subintervals.
在 IB 题目中，你可能需要分析多项式、指数函数和三角函数的单调性。也可能会讨论“非减”或“非增”的情况，此时函数在某些子区间上保持水平。
4. One‑to‑One and Many‑to‑One Functions | 一一函数与多一函数
A function is one‑to‑ passes the horizontal line test: every horizontal line intersects the graph at most once. This property is crucial for an inverse function to exist.
如果一个函数的不同输入总是产生不同的输出，即 f(x₁) = f(x₂) 蕴含 x₁ = x₂，则该函数是一一函数（单射）。其图像通过水平线检验：任意一条水平线与图像最多相交一次。该性质对于反函数的存在至关重要。
Many‑to‑ ctions map two or more different inputs to the same output. For example, f(x) = x² is many‑to‑one because f(2) = 4 and f(– ontal line test. Only by restricting the domain (e.g., x ≥ 0) can we make the function one‑to‑one and thus invertible.
多一函数将两个或多个不同的输入映射到同一个输出。例如 f(x) = x² 是多一函数，因为 f(2) = 4 且 f(–2) = 4。其图像不能通过水平线检验。只有通过限制定义域（如 x ≥ 0），才能使其成为一一函数，从而可逆。
IB exams frequently ask to justify whether a function is one‑to‑one using the derivative sign (strict monotonicity implies injectivity) or by using the horizontal line test on a given graph.
IB 考试经常要求利用导数符号（严格单调性蕴含单射性）或在给定图像上使用水平线检验来论证某个函数是否是一一函数。
5. Inverse Functions | 反函数
If f is a one‑to‑ n R. Graphically, the curve of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.
如果 f 是一一函数，定义域为 D，值域为 R，其反函数 f⁻¹ 的定义域为 R，值域为 D，且对所有 x ∈ D 满足 f⁻¹(f(x)) = x，对所有 y ∈ R 满足 f(f⁻¹(y)) = y。从图像上看，y = f⁻¹(x) 的曲线是 y = f(x) 关于直线 y = x 的反射。
olve for y. For instance, f(x) = 2x + 3 gives y = 2x + 3, swap to x = 2y + 3, solve y = (x – 3)/2, so f⁻¹(x) = (x – 3)/2. Always state the domain of f⁻¹, which is the range of the original function f.
通过代数方法求反函数时，写出 y = f(x)，交换 x 和 y，然后解出 y。例如 f(x) = 2x + 3 给出 y = 2x + 3，交换后 x = 2y + 3，解得 y = (x – 3)/2，因此 f⁻¹(x) = (x – 3)/2。务必标明 f⁻¹ 的定义域，即原函数 f 的值域。
If the original function is not one‑to‑ n restriction must be chosen. For y = x², restricting to x ≥ 0 yields the inverse f⁻¹(x) = √x. IB questions often link domain restriction, monotonicity, and the existence of an inverse.
如果原函数不是一一函数，必须选择适当的定义域限制。对于 y = x²，限制 x ≥ 0 后得到反函数 f⁻¹(x) = √x。IB 题目常常将定义域限制、单调性和反函数的存在性关联起来考查。
6. Composition of Functions | 复合函数
Given two functions f and g, the composite function (f ∘ ange of g is a subset of the domain of f. The order matters: in general, f(g(x)) ≠ g(f(x)).
给定两个函数 f 和 g，复合函数 (f ∘ g)(x) = f(g(x)) 仅在 g 的值域包含于 f 的定义域时才存在。顺序至关重要：一般来说，f(g(x)) ≠ g(f(x))。
For example, let f(x) = √x (domain x ≥ 0) and g(x) = x – 3. Then (f ∘ g)(x) = √x – 3 requires x – 3 ≥ 0 ⇒ omain of the composite function is x ≥ 3, even though g alone has domain all real numbers.
例如，设 f(x) = √x（定义域 x ≥ 0）和 g(x) = x – 3。则 (f ∘ g)(x) = √x – 3 要求 x – 3 ≥ 0 ⇒ x ≥ 3。复合函数的定义域为 x ≥ 3，尽管 g 本身的定义域是所有实数。
complicated function into simpler components is a useful technique. The function h(x) = 1/√2x + 1 can be seen as f(g(x)) with g(x) = 2x + 1 and f(u) = 1/√u. IB exercises often ask to form composites, determine their domains, and deconstruct nested functions.
将复杂函数拆分成基本部分是一种有用的技巧。函数 h(x) = 1/√2x + 1 可看作 f(g(x))，其中 g(x) = 2x + 1，f(u) = 1/√u。IB 练习通常要求构造复合函数、确定其定义域，并拆解嵌套函数。
7. Periodicity | 周期性
mallest such p is called the fundamental period. The most familiar periodic functions are trigonometric: sin x and cos x have period 2π; tan x has period π.
如果存在正的常数 p 使对定义域内所有 x 有 f(x + p) = f(x)，则函数 f 是周期函数。满足该条件的最小正数 p 称为基本周期。最常见的周期函数是三角函数：sin x 和 cos x 的周期为 2π，tan x 的周期为 π。
and minimum values. Transformations such as f(x) = a sin(bx + c) + d change period to 2π/|b|, amplitude to |a|, and shift the graph vertically and horizontally.
周期函数的图像每隔一定长度重复出现。正弦和余弦函数的振幅是最大值与最小值差的一半。变换如 f(x) = a sin(bx + c) + d 将周期变为 2π/|b|，振幅变为 |a|，并使图像沿垂直和水平方向平移。
In IB, periodicity may also appear in the context of modelling waves, tides, or seasonal data. Recognising the period helps predict function values outside the given interval.
在 IB 中，周期性也可能出现在波动、潮汐或季节性数据的建模中。识别周期有助于预测给定区间之外的函数值。
8. Boundedness | 有界性
ounded if it is bounded both above and below. For example, f(x) = sin x is bounded between –1 and 1; g(x) = eˣ is bounded below by 0 but not bounded above.
若存在实数 M 使得对定义域内所有 x 有 f(x) ≤ M，则函数有上界。若存在实数 m 使得 f(x) ≥ m，则有下界。如果同时有上界和下界，则函数有界。例如 f(x) = sin x 在 –1 和 1 之间有界；g(x) = eˣ 有下界 0 但无上界。
ions often exhibit unbounded behaviour near vertical asymptotes. For instance, f(x) = 1/(x – gative values. Boundedness helps in understanding limits and the overall shape of a graph.
有理函数常在垂直渐近线附近表现为无界。例如 f(x) = 1/(x – 1) 在 x = 1 附近无界，可取任意大的正值和负值。有界性有助于理解极限和图像的整体形态。
on like f(x) = x² + 2x + 3 can be written as (x+1)² + 2, showing it is bounded below by 2.
IB 题目有时要求用集合记号表示值域，这直接揭示有界性。像 f(x) = x² + 2x + 3 这样的函数可以写成 (x+1)² + 2，表明它有下界 2。
9. Asymptotes and End Behaviour | 渐近线与末端行为
symptotes occur where the function is undefined due to division by zero, e.g., f(x) = 2/(x – escribe the behaviour as x → ±∞: if limₓ→∞ f(x) = L, then y = L is a horizontal asymptote.
渐近线是函数图像趋近但永远不触及（或仅在无穷远处触及）的直线。垂直渐近线出现在函数因分母为零而无定义的地方，如 f(x) = 2/(x – 3) 在 x = 3 处有垂直渐近线。水平渐近线描述 x → ±∞ 时的行为：若 limₓ→∞ f(x) = L，则 y = L 是水平渐近线。
ique asymptote y = x. Behaviour near asymptotes is essential for accurate graph sketching.
当分子的次数比分母正好高一次时，有理函数会出现斜渐近线。例如 f(x) = (x² + 1)/x 可改写为 x + 1/x，揭示出斜渐近线 y = x。渐近线附近的行为对于准确绘制草图至关重要。
. Always test both sides of a vertical asymptote to see whether the function goes to +∞ or –∞.
IB 考试要求通过长除法或取极限来求渐近线。对于垂直渐近线，务必检验其左右两侧，看函数是趋向 +∞ 还是 –∞。
10. Symmetry and Transformations | 对称性与变换
Beyond even and odd symmetry, functions can exhibit translational or reflective symmetry through combinations of transformations. The standard toolkit of transformations includes vertical translation f(x) + k, horizontal translation f(x – h), vertical stretch a·f(x), horizontal stretch f(bx), and reflections –f(x) or f(–x). These can be applied in sequence, but the order matters: horizontal changes affect the x‑values and must be factored correctly.
除了奇偶对称性，函数还可以通过变换的组合呈现平移或反射对称。标准的变换工具包括垂直平移 f(x) + k、水平平移 f(x – h)、垂直伸缩 a·f(x)、水平伸缩 f(bx)，以及反射 –f(x) 或 f(–x)。这些变换可以依次应用，但顺序很重要：水平变换影响 x 值，必须正确提取公因子。
For a function like h(x) = 3 sin(2x – π) + 1, rewrite as 3 sin[2(x – g mapping of key points (x, y) → (x/2 + π/2, 3y + 1) deepens the grasp of transformations.
对于像 h(x) = 3 sin(2x – π) + 1 这样的函数，改写为 3 sin[2(x – π/2)] + 1。变换步骤为：水平伸长为原来的 1/2 倍，右移 π/2，垂直伸长 3 倍，上移 1 个单位。理解关键点映射 (x, y) → (x/2 + π/2, 3y + 1) 能够加深对变换的掌握。
IB exams frequently ask you to describe the sequence of transformations that produce a new graph from a parent function, and to determine the coordinates of image points after multiple transformations.
IB 考试经常要求描述从母函数生成新图像的一系列变换，并确定经多次变换后的像点坐标。
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